statsmodels.VARMAX against drvarma and drtran¶
Measured, not argued. Everything below was run on the two series that ship with
atsw (IPC_ES.csv, WTI.csv, 2002-01…2019-12), statsmodels 0.14.6.
The short answer: on the same specification the engines agree to four or five decimals, and the interesting differences are elsewhere.
1. The same VAR(1), both engines¶
(Δln WTI, Δln CPI) × 100, 214 observations, VAR(1) with a mean.
statsmodels VARMAX |
drvarma (exact ML) |
|
|---|---|---|
| φ (WTI lag, CPI equation) | 0.020967 | 0.020952 |
| φ (CPI lag, CPI equation) | 0.145772 | 0.144644 |
| φ (WTI lag, WTI equation) | 0.322695 | 0.321768 |
| corr of the innovations | 0.2336 | 0.2313 |
Agreement to ~1e-5 on the pass-through coefficient and ~1e-3 on the rest. The log-likelihoods are not directly comparable (−926.35 against −929.73) because the two use different pre-sample conventions — statsmodels' Kalman filter with its initialisation, drvarma's exact VARMA likelihood — but the estimates land in the same place.
So "which engine is more accurate" is not the question to ask. For a plain VAR on a decent sample, both are right.
2. Where the numbers actually change: the specification¶
The same data, the same engine, one option:
deseason |
φ_WTI,1 | φ_CPI,1 | corr | logL |
|---|---|---|---|---|
None |
0.020952 | 0.144644 | 0.2313 | −929.73 |
"auto" |
0.010421 | 0.209699 | 0.5144 | −731.02 |
"force" |
0.011901 | 0.188451 | 0.5276 | −721.68 |
| published study | 0.0099 | 0.2317 | 0.51 |
Removing the seasonal component halves the estimated pass-through
coefficient and more than doubles the contemporaneous correlation, from
0.23 to 0.51. And deseason="auto" is what reproduces the published analysis —
that is how the study's numbers were obtained, and running without it does not
approximate them, it contradicts them.
Why this matters more than the engine comparison: the study's headline result is a variance decomposition, and its own text says the contemporaneous correlation "is the main driver of the differences in variance decompositions under alternative orderings". A correlation of 0.23 and one of 0.51 give different answers to the question the study exists to ask.
statsmodels.VARMAX has no seasonal handling at all. Neither a deseason
option nor seasonal differencing: SARIMAX has them, VARMAX does not. So the
0.23 row is what a careful user gets by default, and getting the 0.51 row
requires knowing to pre-process — and pre-processing the two series consistently
by hand.
3. What each can express¶
This is where the packages genuinely differ, and it is a matter of scope rather than quality.
statsmodels |
drvarma / drtran |
|
|---|---|---|
| VAR, VARMA | yes | yes (exact ML) |
| exogenous regressors | yes, as contemporaneous regression coefficients | — |
rational transfer ω(B)/δ(B)·B^b |
no | drtran: yes, with delay and denominator |
| networks of transfers (a DAG) | no | drtran: yes |
| seasonality frequency by frequency | no | ifadf: yes (the MEG/HSM class) |
| constraints: fixed, shared, products | no | drtran: a .cns table |
| identification (what orders, what λ, what d) | you specify | art: guided, node by node |
The transfer row is the substantive one. VARMAX(exog=…) regresses on the
exogenous variable contemporaneously; expressing "oil affects prices with a
delay of b and a geometric decay 1/δ(B)" means building the lags yourself
and losing the parsimony that made the rational form worth having. drtran
estimates b, ω(B) and δ(B) jointly by exact ML, reports the gain ν(1)
and the mean lag, and can chain several such links in a DAG.
Conversely statsmodels covers a great deal this suite does not — GARCH-type
state space, unobserved components, regime switching, a vast test library — and
is the right tool for most of them.
3b. Is VARMAX an alternative to drtran? No — but ARDL is the near miss¶
VARMAX does accept exogenous variables, so the question is fair. Its own documentation writes the model as
y_t = A(t) + A_1 y_{t-1} + … + A_p y_{t-p} + B x_t + ε_t + M_1 ε_{t-1} + …
B x_t — contemporaneous, and nothing else. No lags of x, no polynomial.
As a transfer function that is the degenerate case ν(B) = ω₀ with b = 0,
r = 0, s = 0: a regression on x with VARMA errors. You can add lags by
building extra columns of exog by hand, and then you have a free coefficient
per lag and no denominator — which is precisely the parsimony the rational form
exists to buy.
The real analogue in statsmodels is statsmodels.tsa.ardl.ARDL, and it is
genuinely a rational transfer. Written out,
φ(B) y_t = c + β(B) x_t + ε_t ⟹ y_t = [β(B)/φ(B)] x_t + [1/φ(B)] ε_t
so the input does pass through a rational filter. Against Box-Jenkins:
y_t = [ω(B)/δ(B)] B^b x_t + [θ(B)/φ(B)] a_t
The difference is one polynomial doing two jobs. In ARDL the transfer's
denominator IS the noise's AR — one φ(B) for both — and there is no MA in the
noise at all. In Box-Jenkins δ(B) and φ(B)/θ(B) are specified and estimated
separately, because there is no reason the dynamics through which oil reaches
prices should be the dynamics of what is left over.
An ARDL can of course approximate a Box-Jenkins transfer by raising its order until one polynomial mimics both. You pay in parameters, and you lose the reading: you can no longer say "this is the transfer's decay and that is the noise's", which is the sentence a transfer model is built to be able to say.
Measured on the example series (Δln × 100, same construction as §1):
| long-run multiplier | note | |
|---|---|---|
ARDL(1,1) |
0.037283 | includes the contemporaneous dwti.L0 = 0.0151 |
drvarma VAR(1), CPI equation |
0.024495 | reduced form has no contemporaneous term |
They differ partly because they are different models — the ARDL has a contemporaneous term the VAR's reduced form cannot have — which is itself the point: which dynamic assumptions you can state is what separates these tools, not how well they optimise.
What ARDL does not offer, and drtran does:
- a delay
bas a parameter of the specification rather than an accident of which lag columns you included; - a separate noise ARMA, so the transfer's decay and the residual dynamics are not forced to share a polynomial;
- identification by prewhitening and the cross-correlation, which reads
(b, r, s)off a correlogram; ARDL selects by information criteria over a grid (ardl_select_order); - networks — several inputs and outputs as a DAG;
- and exact ML for the whole system at once; ARDL is conditional least squares.
And a search of statsmodels.tsa for "transfer function" or "rational lag"
returns nothing. The Box-Jenkins transfer function, with ω(B)/δ(B)B^b and
an independently specified noise, is not implemented there.
4. What to take from this¶
- Do not choose on the engine. Two independent implementations of exact ML and of the Kalman filter agree to 1e-5 on the same model. If two programs disagree materially, suspect the specification first — and this comparison is a worked example of that suspicion being right.
- Choose on what the tool lets you say. A rational transfer function, a per-frequency seasonal decision or a constrained parameter table are either expressible or they are not.
- And choose on what the tool makes hard to get wrong. The 0.23-versus-0.51 row is not a bug in anything. It is a modelling decision that one package asks about and the other leaves entirely to the user — and the study's conclusion turns on it.
Reproduce the table yourself: the two series ship with
atsw(atsw.example_path("IPC_ES.csv")), and every figure above comes from a twenty-line script.